Semiclassical theory of transport in a random magnetic field

We present a systematic description of the semiclassical kinetics of two-dimensional fermions in a smoothly varying inhomogeneous magnetic field B(r). The nature of the transport depends crucially on both the strength B0 of the random component of B(r) and its mean value ¯B. For ¯B=0, the governing parameter is α=d/R0, where d is the correlation length of disorder and R0 is the Larmor radius in the field B0. While for α≪1 the Drude theory applies, at α≫1 most particles drift adiabatically along closed contours and are localized in the adiabatic approximation. The conductivity is then determined by a special class of trajectories, the “snake states,” which percolate by scattering at saddle points of B(r) where the adiabaticity of their motion breaks down. The external field ¯B also suppresses the diffusion by creating a percolation network of drifting cyclotron orbits. This kind of percolation is due only to a weak violation of the adiabaticity of the cyclotron rotation, yielding an exponentially fast drop of the conductivity at large ¯B. In the regime α≫1, the crossover between the snake-state percolation and the percolation of the drift orbits with increasing ¯B has the character of a phase transition (localization of the snake states) smeared exponentially weakly by nonadiabatic effects. The ac conductivity also reflects the dynamical properties of particles moving on the fractal percolation network. In particular, it has a sharp kink at zero frequency and falls off exponentially at higher frequencies. We also discuss the nature of the quantum magneto-oscillations. Detailed numerical studies confirm the analytical findings. The shape of the magnetoresistivity at α∼1 is in good agreement with experimental data in the fractional quantum Hall regime near half filling